Adaptive racing ranking-based immune optimization approach solving multi-objective expected value programming

Abstract

This work investigates a bio-inspired adaptive sampling immune optimization approach to solve a general kind of nonlinear multi-objective expected value programming without any prior noise distribution. A useful lower bound estimate is first developed to restrict the sample sizes of random variables. Second, an adaptive racing ranking scheme is designed to identify those valuable individuals in the current population, by which high-quality individuals in the process of solution search can acquire large sample sizes and high importance levels. Thereafter, an immune-inspired optimization approach is constructed to seek $\varepsilon$-Pareto optimal solutions, depending on a novel polymerization degree model. Comparative experiments have validated that the proposed approach with high efficiency is a competitive optimizer.

Appendices

Appendix 1

Proof Lemma 1

In the case of ${{\varvec{x}}}\in A_\varepsilon$, there exists at least a subscription $j_0$ satisfying that

$$\begin{aligned} E[f_{j_0 } ({{\varvec{x}}},{\varvec{\xi } })]+\varepsilon <E[f_{j_0 } ({{\varvec{y}}},{\varvec{\xi } })]. \end{aligned}$$

(18)

Hence, once${\hat{{f}}}_{j_0 } ({{\varvec{y}}})+\varepsilon \le {\hat{{f}}}_{j_0 } ({{\varvec{x}}})$, one can imply that

$$\begin{aligned} {\hat{{f}}}_{j_0 } ({{\varvec{y}}})+\varepsilon <E[f_{j_0 } ({{\varvec{y}}},{\varvec{\xi } })]-\frac{\varepsilon }{2} \end{aligned}$$

or

$$\begin{aligned} {\hat{{f}}}_{j_0 } ({{\varvec{x}}})>E[f_{j_0 } ({{\varvec{x}}},\xi )]+\frac{\varepsilon }{2}. \end{aligned}$$

Otherwise, by Eq. (18) we have that

$$\begin{aligned} {\hat{{f}}}_{j_0 } ({{\varvec{y}}})+\varepsilon \ge E[f_{j_0 } ({{\varvec{y}}},{\varvec{\xi } })]-\frac{\varepsilon }{2}>E[f_{j_0 } ({{\varvec{x}}},{\varvec{\xi } })]+\frac{\varepsilon }{2}\ge {\hat{{f}}}_{j_0 } ({{\varvec{x}}}). \end{aligned}$$

(19)

This yields contradiction. Consequently,

$$\begin{aligned}&\Pr \left\{ {{\hat{{f}}}_{j_0 } (y)+\varepsilon \le {\hat{{f}}}_{j_0 } (x)} \right\} \le \Pr \{{\hat{{f}}}_{j_0 } (y)<E[f_{j_0 } (y,\xi )] \nonumber \\&\quad -\frac{3\varepsilon }{2}\}+\Pr \{{\hat{{f}}}_{j_0 } (x)>E[f_{j_0 } (x,{\varvec{\xi }})]+\frac{\varepsilon }{2}\}. \end{aligned}$$

(20)

Further, it follows from Eq. (20) and Theorem 1 that

$$\begin{aligned} \Pr \left\{ {{\hat{{f}}}_{j_0 } ({{\varvec{y}}})+\varepsilon \le {\hat{{f}}}_{j_0 } ({{\varvec{x}}})} \right\} \le 2e^{-mc\varepsilon ^{2}}. \end{aligned}$$

(21)

Thus,

$$\begin{aligned}&\Pr \left\{ {{{\varvec{y}}}\prec _{\hat{{\varepsilon }}} {{\varvec{x}}}} \right\} \le \prod _{j=1}^q {\Pr \left\{ {{\hat{{f}}}_j ({{\varvec{y}}}) +\varepsilon \le {\hat{{f}}}_j ({{\varvec{x}}})} \right\} } \\&\quad \le \Pr \left\{ {{\hat{{f}}}_{j_0 } ({{\varvec{y}}})+\varepsilon \le {\hat{{f}}}_{j_0 } ({{\varvec{x}}})} \right\} \le 2e^{-mc\varepsilon ^{2}}.\nonumber \end{aligned}$$

(22)

The proof is completed. $\square$

Proof of Theorem 2

For a given ${{\varvec{x}}}\in A_\varepsilon$, if ${{\varvec{x}}}\notin {\hat{{A}}}_\varepsilon$, there exists ${{\varvec{y}}}\in {\hat{{A}}}_\varepsilon$ such that ${{\varvec{y}}}\prec _{\hat{{\varepsilon }}} {{\varvec{x}}}$. Hence,

$$\begin{aligned} \Pr \{{{\varvec{x}}}\notin {\hat{{A}}}_\varepsilon \}\le & {} \Pr \{\exists {{\varvec{y}}}\in {\hat{{A}}}_\varepsilon ,s.t.{{\varvec{y}}}\prec _{\hat{{\varepsilon }}} {{\varvec{x}}}\}\nonumber \\\le & {} \sum _{{{\varvec{y}}}\in {\hat{{A}}}_\varepsilon } {\Pr \{{{\varvec{y}}}\prec _{\hat{{\varepsilon }}} {{\varvec{x}}}\}} , \end{aligned}$$

(23)

and accordingly,

$$\begin{aligned} \Pr \{A_\varepsilon \not \subset {\hat{{A}}}_\varepsilon \}= & {} \Pr \{\exists {{\varvec{x}}}\in A_\varepsilon s.t.{{\varvec{x}}}\notin {\hat{{A}}}_\varepsilon \} \\\le & {} \Pr \{\exists {{\varvec{x}}}\in A_\varepsilon ,{{\varvec{y}}}\in {\hat{{A}}}_\varepsilon ,s.t.{{\varvec{y}}}\prec _{\hat{{\varepsilon }}} {{\varvec{x}}}\} \nonumber \\\le & {} \sum _{{{\varvec{x}}}\in A_\varepsilon } {\sum _{{{\varvec{y}}}\in {\hat{{A}}}_\varepsilon } {\Pr \{{{\varvec{y}}}} } \prec _{\hat{{\varepsilon }}} {{\varvec{x}}}\}. \nonumber \end{aligned}$$

(24)

Therefore, as related to Lemma 1 and Eq. (24), we can obtain that

$$\begin{aligned} \Pr \{A_\varepsilon \subseteq {\hat{{A}}}_\varepsilon \}=1-\Pr \{A_\varepsilon \not \subset {\hat{{A}}}_\varepsilon \}\ge 1-2N^{2}e^{-mc\varepsilon ^{2}}. \end{aligned}$$

(25)

This finishes the proof. $\square$

The proof of Theorem 3

In the case of $x\in A_\varepsilon$, we know that

$$\begin{aligned} \Pr \{{{\varvec{x}}}\notin {\hat{{A}}}_\varepsilon \}=\Pr \{\exists {{\varvec{y}}}\in {\hat{{A}}}_\varepsilon ,s.t. {{{\varvec{y}}}\prec _{\hat{{\varepsilon }}} {{\varvec{x}}}}\} \le \sum _{{{\varvec{y}}}\in {\hat{{A}}}_\varepsilon } {\Pr \left\{ {{{\varvec{y}}}\prec _{\hat{{\varepsilon }}} {{\varvec{x}}}} \right\} } . \end{aligned}$$

(26)

By Lemma 1, Eq. (21) implies that

$$\begin{aligned} \Pr \{{{\varvec{x}}}\notin {\hat{{A}}}_\varepsilon \}\le 2|{\hat{{A}}}_\varepsilon |e^{-mc\varepsilon ^{2}}. \end{aligned}$$

(27)

Subsequently, we can easily obtain that

$$\begin{aligned} \Pr \{{\hat{{A}}}_\varepsilon \cap A_\varepsilon =\emptyset \}\le \prod _{x\in A_\varepsilon } {\Pr \{x\notin {\hat{{A}}}_\varepsilon \}} \le 2Ne^{-mc\varepsilon ^{2}}. \end{aligned}$$

(28)

Thereby, the conclusion is true. $\square$

The proof of Theorem 3

In Step 4, those $\beta$-dominated and redundant B cells in $\hbox {M}_{set}$ will be eliminated after copying $\hbox {B}_{n}$ into $\hbox {M}_{set}$, for which the computational complexity is $\hbox {O}({\vert }\hbox {M}_{set}{\vert }\log {\vert }\hbox {M}_{set}{\vert })$. In particular, it is possible that the number of B cells in M$_{set}$ is beyond ($N+M)$. Thus, in the worst case we can assert that the complexity of Step 4 is $\hbox {O}((N+M) \log (N+M))$. Step 5 needs to calculate PDM values of elements in $A_{n}$; more precisely, we need to calculate their crowding distances with at most $N\hbox {C}_{\max }\log N\hbox {C}_{\max }$ times and their values on S with $(N\hbox {C}_{\max }+1)N\hbox {C}_{\max }/2$ times. Step 8 executes mutation with at most $Np\hbox {C}_{\max }$ times. In Step 9, the size of $\hbox {B}_{n}\cup \hbox {E}_{n}$ is at most $\hbox {NC}_{\max }$. Thus, when ARRA is enforced on $\hbox {B}_{n}\cup \hbox {E}_{n}$, the complexity in the worst case is $O(NC_{\max } (M_2 +\log NC_{\max } ))$ by means of the computational complexity in Sect. 5.1. Summarily, MEIOA’s computational complexity in the worst case is decided by

$$\begin{aligned} O_c= & {} O\left( {(N+M)\log (N+M)} \right) +O\left( {NpC_{\max } } \right) \\&+O\left( {NC_{\max } (M_2 +\log NC_{\max } )} \right) +O((NC_{\max } )^{2}) \\= & {} O\left( (N+M)\log (N+M)+NC_{\max } (M_2 +p\right. \\&\left. +NC_{\max } ) \right) . \end{aligned}$$

Therefore, the conclusion is right. $\square$

Appendix 2

See Table 7.

Table 7 The first test suite (Van Veldhuizen 1999)

Table 8 The second test suite (Zitzler et al. 2000)

Appendix 3

See Table 8.